### Abstract

In evolutionary game theory, the effect of public goods like diffusible molecules has been modelled using linear, concave, sigmoid and step functions. The observation that biological systems are often sigmoid input–output functions, as described by the Hill equation, suggests that a sigmoid function is more realistic. The Michaelis–Menten model of enzyme kinetics, however, predicts a concave function, and while mechanistic explanations of sigmoid kinetics exist, we lack an adaptive explanation: what is the evolutionary advantage of a sigmoid benefit function? We analyse public goods games in which the shape of the benefit function can evolve, in order to determine the optimal and evolutionarily stable Hill coefficients. We find that, while the dynamics depends on whether output is controlled at the level of the individual or the population, intermediate or high Hill coefficients often evolve, leading to sigmoid input–output functions that for some parameters are so steep to resemble a step function (an on–off switch). Our results suggest that, even when the shape of the benefit function is unknown, biological public goods should be modelled using a sigmoid or step function rather than a linear or concave function.

Original language | English |
---|---|

Pages (from-to) | 73-82 |

Number of pages | 10 |

Journal | Journal of Theoretical Biology |

Volume | 406 |

DOIs | |

Publication status | Published - Oct 7 2016 |

### Fingerprint

### Keywords

- Cooperation
- Enzyme kinetics
- Game theory
- Hill equation
- Mechanism design
- Michaelis–Menten
- Public goods

### ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability
- Modelling and Simulation
- Agricultural and Biological Sciences(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Medicine(all)

### Cite this

**Evolution of optimal Hill coefficients in nonlinear public goods games.** / Archetti, Marco; Scheuring, I.

Research output: Contribution to journal › Article

*Journal of Theoretical Biology*, vol. 406, pp. 73-82. https://doi.org/10.1016/j.jtbi.2016.06.030

}

TY - JOUR

T1 - Evolution of optimal Hill coefficients in nonlinear public goods games

AU - Archetti, Marco

AU - Scheuring, I.

PY - 2016/10/7

Y1 - 2016/10/7

N2 - In evolutionary game theory, the effect of public goods like diffusible molecules has been modelled using linear, concave, sigmoid and step functions. The observation that biological systems are often sigmoid input–output functions, as described by the Hill equation, suggests that a sigmoid function is more realistic. The Michaelis–Menten model of enzyme kinetics, however, predicts a concave function, and while mechanistic explanations of sigmoid kinetics exist, we lack an adaptive explanation: what is the evolutionary advantage of a sigmoid benefit function? We analyse public goods games in which the shape of the benefit function can evolve, in order to determine the optimal and evolutionarily stable Hill coefficients. We find that, while the dynamics depends on whether output is controlled at the level of the individual or the population, intermediate or high Hill coefficients often evolve, leading to sigmoid input–output functions that for some parameters are so steep to resemble a step function (an on–off switch). Our results suggest that, even when the shape of the benefit function is unknown, biological public goods should be modelled using a sigmoid or step function rather than a linear or concave function.

AB - In evolutionary game theory, the effect of public goods like diffusible molecules has been modelled using linear, concave, sigmoid and step functions. The observation that biological systems are often sigmoid input–output functions, as described by the Hill equation, suggests that a sigmoid function is more realistic. The Michaelis–Menten model of enzyme kinetics, however, predicts a concave function, and while mechanistic explanations of sigmoid kinetics exist, we lack an adaptive explanation: what is the evolutionary advantage of a sigmoid benefit function? We analyse public goods games in which the shape of the benefit function can evolve, in order to determine the optimal and evolutionarily stable Hill coefficients. We find that, while the dynamics depends on whether output is controlled at the level of the individual or the population, intermediate or high Hill coefficients often evolve, leading to sigmoid input–output functions that for some parameters are so steep to resemble a step function (an on–off switch). Our results suggest that, even when the shape of the benefit function is unknown, biological public goods should be modelled using a sigmoid or step function rather than a linear or concave function.

KW - Cooperation

KW - Enzyme kinetics

KW - Game theory

KW - Hill equation

KW - Mechanism design

KW - Michaelis–Menten

KW - Public goods

UR - http://www.scopus.com/inward/record.url?scp=84978880203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978880203&partnerID=8YFLogxK

U2 - 10.1016/j.jtbi.2016.06.030

DO - 10.1016/j.jtbi.2016.06.030

M3 - Article

C2 - 27343626

AN - SCOPUS:84978880203

VL - 406

SP - 73

EP - 82

JO - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

ER -